# In differential equations, do transient state and steady state always go hand in hand?

Do transient state and steady state go hand in hand in differential equations? Meaning, if there is a transient state, is there always going to be a steady state?

Also, if there are neither of them, then is it always pure resonance?

Could a DE not have any of those states?

• Do you have access to the book by Tenenbaum and Pollard? – Amzoti Dec 18 '13 at 1:00
• @Amzoti No I don't. – Joshua Ree Dec 18 '13 at 5:00
• @JoMo: Did the answer resolve your question? – Amzoti Dec 19 '13 at 2:37

Lets say we have a spring with an external force.

We solve the second order ODE and arrive at:

$$x(t) = \dfrac{e^{-4 t}}{9} [3 \cos(4 \sqrt{3} t) - 11 \sqrt{3}) \sin(4 \sqrt{3} t)] + 2 \sin(8 t) \\ y(t) = 2 \sin(8t)$$

A plot of these functions is:

You can see that the $e^{-4t}$ becomes negligible when $t$ is large. The steady state solution is shown in blue and the steady-state plus transient is shown in red.

These transient terms in the solution, when they are significant, are sometimes called the transient solutions. In many physical problems, the transient solution is the least important part. However, there are cases where it is of major importance.

When the transient terms are negligible, only the $2\sin(8t)$ remains. This is called the steady-state term or steady-state solution, since it indicates the behavior of the system when conditions have become steady. You can see in the graph, that the steady-state solution is periodic and has the same period as that of the applied external force (in this case, that was, in this example, $F(t) = 24 \cos(8t)$).

Electrical circuits with resistors, capacitors and inductors (like an RLC circuit) look at transient and steady state quite a bit, so you might want to explore those.

Look at the book by Tenenbaum and Pollard for more examples regarding the damped and undamped frequency and why these things can become critically important.

• Deserves an UP! +1 – Namaste Dec 19 '13 at 0:07

I stick strictly to your question.
Of course you are speaking of a second order linear (ordinary) differential equation with constant coefficients $$x''+ \alpha x'+\beta x=f(t)$$The theory says that its general solution is $$x=x_h+x_p$$ where $x_h$ is the general solution of the associated homogeneous equation and $x_p$ is any particular solution.
It is a fact that $$\lim_{t\rightarrow\infty}x_h=0$$ if and only if $\,\alpha>0$ and $\beta>0$.
In that case, with reference to the typical applications to physics where $t$ is time, one speaks of $x_h$ as the transient solution and of $x_p$ as the steady state solution.
If $\alpha=0$ , $\beta>0$ and in the Fouries series of $f$ (supposed periodic) there is a term with frequency equal to $\sqrt \beta$, then one has pure resonance (no transient solution, because $\alpha=0$, so speaking of a steady state solution is nonsense).

• This is a wonderful explanation, thank-you. – jake mckenzie Sep 13 '16 at 21:22